Polycyclic-by-finite groups admit a bounded-degree polynomial structure
type: Item
instance of: scholarly article
title: Polycyclic-by-finite groups admit a bounded-degree polynomial structure
zbMATH Open document ID: 0867.20031
author: Karel Dekimpe
author: Paul Igodt
published in: Inventiones Mathematicae
publication date: 1997-06-22T00:00:00Z
review text: For a polycyclic-by-finite group \(\Gamma\), of Hirsch length \(h\), an affine (resp. polynomial) structure is a representation of \(\Gamma\) into \(\text{Aff}(\mathbb{R}^h)\) (resp. \(P(\mathbb{R}^h)\), the group of polynomial diffeomorphisms) letting \(\Gamma\) act properly discontinuous on \(\mathbb{R}^h\). Recently it was shown by counter-examples that there exist groups \(\Gamma\) (even nilpotent ones) which do not admit an affine structure, thus giving a negative answer to a long-standing question of John Milnor. We prove that every polycyclic-by-finite group \(\Gamma\) admits a polynomial structure, which moreover appears to be of a special (``simple'') type (called ``canonical'') and, as a consequence of this, consists entirely of polynomials of a bounded degree. The construction of this polynomial structure is a special case of an iterated Seifert fiber space construction, which can be achieved here because of a very strong and surprising cohomology vanishing theorem.
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 57S05
zbMATH DE Number: 1000569
zbMATH Keywords: polycyclic-by-finite groups
zbMATH Keywords: Hirsch length
zbMATH Keywords: groups of polynomial diffeomorphisms
zbMATH Keywords: cohomology vanishing theorem
MaRDI profile type: Publication
full work available at URL: s002220050160
OpenAlex ID: W1977343094
DOI: 10.1007/S002220050160
description: scientific article; zbMATH DE number 1000569